Tail inequalities for restricted classes of discrete random variables

TL;DR

This paper derives new tail inequalities for specific classes of discrete random variables, providing bounds based on their distributional properties such as monotonicity and unimodality.

Contribution

It introduces novel tail bounds for discrete variables with monotone decreasing probabilities and unimodal distributions, extending classical inequalities.

Findings

Bound for discrete variables with decreasing probabilities: P(X ≤ a) ≤ E[X]/(2a - 1)

Bound for unimodal variables: P(|W - E[W]| ≥ a) ≤ (Var(W) + 1/12) / (2(a - 1/2)^2)

Applicable to variables with finite second moments and specific distributional constraints.

Abstract

Let $X$ be an integrable discrete random variable over $\{0, 1, 2, \ldots\}$ with $\mathbb{P}(X = i + 1) \leq \mathbb{P}(X = i)$ for all $i$. Then for any integer $a \geq 1$, $\mathbb{P}(X \leq a) \leq \mathbb{E}[X] / (2a - 1)$. Let $W$ be an discrete random variable over $\{\ldots, -2, -1, 0, 1, 2, \ldots\}$ with finite second moment where the $\mathbb{P}(W = i)$ values are unimodal. Then $\mathbb{P}(|W - \mathbb{E}[W]| \geq a) \leq (\mathbb{V}(W) + 1 / 12) / (2(a - 1 / 2)^2)$.